Design of 3-D Periodic Metamaterials for ...

910

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 58, NO. 4, APRIL 2010

Design of 3-D Periodic Metamaterials for Electromagnetic Properties Shiwei Zhou, Wei Li, and Qing Li

Abstract—This paper presents a new homogenization formula to compute the effective electromagnetic properties for periodic metamaterials. Numerical examples showed that the effective permittivity and permeability of the composites with cubic inclusions, formerly known to have the lowest permittivity, are closer to the Hashin–Strikman bounds than those obtained from other methods. To tailor the specific effective properties, an inverse homogenization procedure is proposed within the framework of vector wave equations. Some novel metamaterial microstructures with a range of specific effective permittivity and/or permeability are obtained. By maximizing the permittivity and permeability at the same time, a structure with minimal surface area (the mean curvature of the surface equals zero everywhere), namely, the well-known Schwarz primitive structure, is obtained. Similarly to the nano-spheres (dielectric spheres covered by plasmonic shells) with negative refraction, we generalize the Schwarz primitive structure and its analogy (e.g., those with a constant mean curvature surface) to one class of chiral composites by embedding one of these structures with smaller volume fraction (nonmagnetic inclusive cores) into another with large volume fraction (metal shell). Such composites have potential to provide better behaviors because they can best utilize different components. The anisotropic composites and multiple solutions to the inverse homogenization are also illustrated. Index Terms—Effective permittivity and permeability, inverse homogenization method, metamaterials, Schwarz primitive structure.

I. INTRODUCTION S THE sizes of metal–dielectric particles periodically suspended in free space or hosted in other homogenized mediums are smaller than a quarter of the wavelengths of Hertzian waves, the electromagnetic fields are largely influenced by the size effects like surface plasmon and mutual interactions between adjacent particles [1]–[3]. Such novel composites are named metamaterials because they could offer extraordinary electromagnetic properties unavailable in nature [1]–[4]. Being one of the main materials with either negative permittivity or negative permeability or both, metamaterials are expected to provide a new arena for many novel applications ranging from biosensors to photonic devices [1], [3]–[9]. The characterizations of metamaterials are popular topics recently. In addition to experimental approaches, various nu-

A

Manuscript received December 11, 2008. First published March 15, 2010; current version published April 14, 2010. This work was supported by the Australian Research Council. The authors are with the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney N.S.W. 2006, Australia (e-mail: [email protected]; [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMTT.2010.2042845

merical methods have been developed to calculate the effective properties for metamaterials. In 1999, Pendry et al. [2] first obtained the uniaxial negative permeability for metallic cylinders with regularly shaped (circle and split ring) cross sections by considering the resonance resulted from internal capacitance and inductance. After that, the effective permittivity of the lattice, constructed by complexly shaped conductive inclusions suspended in a host medium, was computed using the moment-method-based technique [8]. It was shown that, by applying unit potential difference and setting periodic boundary conditions onto the opposite boundaries in a square or cubic domain, the effective-medium-based method [10], [11] can compute the effective electromagnetic properties for normal composites as well as metamaterials. More recently, the homogenization technique [12], [13], widely used for computing the elasticity tensors [14], was adopted to derive the effective electromagnetic properties by Quchetto et al. [5]. The homogenization method benefits the characterization of metamaterials from at least two aspects. First, the fact that many metamaterials are fabricated by periodically repeated representative volume elements (base cells or resonators) allows restricting the analysis to a small representative domain rather than the entire composite material. Second, the size of the base cell is relatively small, which well meets the multiscale requirement in the homogenization technique. The most striking advantage of the homogenization method lies in that it is possible to seek some optimal configurations because it is applicable to any arbitrarily shaped base cell. Unlike the homogenization method in [5], this paper proposes a different homogenization formulation, in which the characteristic fields used in the integral of the electromagnetic properties are in a vectorial form rather than a scalar potential. Numerical tests indicate that this appears to be more suitable to commonly used composites, whose inclusions are suspended in either a free space or a hosted homogenous medium. This new homogenization technique will be further used in an inverse procedure [15] in this study to tailor the effective properties for some special engineering needs. Compared with traditional size optimization, where the ratio of the width of the cubic inclusion to the corner radius was optimized [8], the inverse homogenization method is more versatile and allows us to optimize the size, shape, and topology of the structures simultaneously. To implement the inverse homogenization in this study, the design domain (base cell) is modeled by using the finite-element method (FEM). The volume fraction of the dielectric or metal phase (i.e., relative density) in each element is taken as the design variable. As a design objective, the squared mismatches between the target and effective values are minimized by using the method of moving asymptotes (MMA) [16]. To drive this gradient-based method here, the sensitivity of the effective values

0018-9480/$26.00 © 2010 IEEE Authorized licensed use limited to: UNIVERSITY OF SYDNEY. Downloaded on April 21,2010 at 05:14:43 UTC from IEEE Xplore. Restrictions apply.

ZHOU et al.: DESIGN OF 3-D PERIODIC METAMATERIALS FOR ELECTROMAGNETIC PROPERTIES

with respect to the design variables is derived by using the adjoint variable method [17]. II. HOMOGENIZATION FORMULATION and , respectively, The electric and magnetic fields, are governed by time-varying (with sinusoidal excitation ) Maxwell’s equations given by (1a) (1b) where , is the curl operator, and the excitation frequency. The electric flux density , magnetic flux density , and the source of electric current , are dependent on the macroscopic properties of media in terms of the constitutive , , and , in which the conrelations stitutive parameters , , and are named as the permittivity, permeability, and conductivity, respectively. These nondispersive (frequency-independent) properties are scalar values when the media are isotropic. However, for anisotropic cases in this paper, they are in a tensor form expressed in the 3 3 matrices. The inhomogeneous vector wave equations can be obtained by eliminating and on the right-hand sides of (1a) and (1b), respectively, given by (2a) (2b)

911

with respect to the macroscopic When derivating coordinate via the chain rule, the derivative across these two scales is obtained as (6) By replacing the electromagnetic fields and corresponding curl operators in (4) with (5) and (6), respectively, and extracting the terms, one can generate the following equilibrium equation:

(7) and are the curl operator matrices in the two difwhere ferent scales, respectively. According to the convergence in the homogenization theory [12]–[14], one can have (8)

where

is the volume of the base cell, and then (7) becomes

Equation (2) can be generalized into a short form as (3) where

, ,

, and . Multiplying a test function on both sides of (3) and integrating by parts according to the relationship with the surface integration of , one can derive the weak variational form of the vector wave equations as

(9) which leads to the following two equilibrium equations:

(4) and denotes the external where surface of the domain occupied by the entire composite. Assuming that the composite is constructed by periodically repeated base cells, the electromagnetic fields can be asymptotically approximated by a polynomial as

(10)

(5) Except for the first term, which only depends on the global (macro) position , the rest are related to both within the global and local (micro) positions the base cell . The contribution of the different hierarchies to the electromagnetic fields is weighted by a small factor of in different exponential orders, which links the . In accordance macro and micro coordinate systems by with the homogenization theory [12]–[14], the two-level hierarchy of (5) provides an approximation to the electromagnetic fields within an acceptable error.

(11) By assuming operator to the both sides, one obtains

and applying the curl (12)

Putting (12) into (11) and eliminating

, we obtain

Authorized licensed use limited to: UNIVERSITY OF SYDNEY. Downloaded on April 21,2010 at 05:14:43 UTC from IEEE Xplore. Restrictions apply.

(13)

912


with identity . Equation (13) can be expressed in a weak form as (14) with bilinear term and linear term into (10) for

Like other algorithms, e.g., evolutionary structural optimization [25], [26], the method of moving asymptotes [16] needs to define how sensitive the objective function is to a perturbation in the design variables. To conduct sensitivity analysis, the first is given by derivative of the cost function

. Substituting

(18)

yields

where the subscript denotes the derivative with respect to the relative density and (19) (15) Comparing (15) with the weak form of the Maxwell’s equation (4), the effective property matrix is thus defined as

The adjoint variable method [17] is used to solve via deriving an adjoint equation as (20)

(16)

III. INVERSE HOMOGENIZATION METHOD FOR THE DESIGN OF METAMATERIALS The formulation in (16) exhibits that three parameters contribute to the effective property tensor. The first two are the physical properties (i.e., permittivities , and permeabilities , ) and the volume fractions ( and ) of the constituent phases. The third is related to the distribution of these compositional materials within the base cell. In our cases, the basic constituent phases and their volume fractions are given, thus the way to tailor the effective properties becomes how to allocate the constituent materials within the base cell properly. Such an effort has been successfully made for various mechanical properties, e.g., negative Poisson’s ratio [18], by means of the inverse homogenization method. This has led, in the first instance, to an understanding of how engineered composites can attain some extraordinary properties that are not readily available in nature [19]–[21]. More recently, further studies concluded that the constituent phases separated by the interfaces with minimum surface area have single or multiple superior physical properties [20]–[23]. The key idea to the inverse homogenization method is that are forced to converge to their targets the effective values in a way that the square of their mismatches is minimized. Thus, the optimization problem is mathematically expressed as

Equation (20) has the same form as (14) except for the negative sign on the right-hand side. Thus, the characteristic equation . (14) is self-adjoint and the adjoint variable is given by By differentiating both sides of the characteristic equation (14) with respect to , one can obtain the following equation: (21) As both

and belong to the same test functional space (periodic Soblev space), substitution of with in (21) yields (22)

Similarly, the substitution of (20) leads to

with

in the adjoint equation (23)

Taking the symmetric bilinear term in (22) and (23), one obtains (24)

Likewise, as both and belong to the same test functional space, one can swap them in (19) and obtain the sensitivity as

(17) where is the local volume fraction (or relative density) of phase and the 1. To avoid intermediate relative densities checkerboard pattern (different phases presents themselves in an alternative fashion), the nonlinear diffusion technique [24], originally used in structural topology optimization, is adopted (the in this procedure. It is noted that the volume constraint volume fraction occupied by Phase 1 within the design domain or base cell) is considered as a constraint in the optimization.

(25)

IV. NUMERICAL IMPLEMENTATION AND RESULTS Based upon the vector wave equations, the characteristic fields, namely the solutions to (14), should be divergence-free.



913

To take this mathematical property into account, a penalty algorithm, originally used in the magnetostatic problem [27], is introduced, leading to a variation of (14) given by (26) and an arbiwith bilinear term trary penalty factor . It is obvious that, when the test function , the second term on the left-hand side of (26) beis . According to the variational principle, comes the solutions to (26) are the stationary point of the total potential , energy thus naturally leading to divergence-free solutions. Within the inverse homogenization framework, the local is assumed to be isotropic and a constant property matrix in each element centered at point . During the optimization process, the relative density might not be necessarily 0 or 1. Hence, we need to evaluate the local properties in such intermediate elements. Since the Hashin–Strikman (HS) bounds [28] provide the lower and upper limits to most physical properties, these two bounds are adopted as the interpolation schemes to calculate the local properties in this paper. The determination of the appropriate bound depends on the target properties wanted. For instance, if the target is closer to the lower HS bound, the upper HS bound should be chosen and vice versa [20], [29]. In addition to the HS bound-based interpolation schemes, the arithmetic bound [30] with an exponential penalty is also applicable. The effect of the shape of the inclusions on the effective permittivity was systematically studied by Whites and Wu by using the moment-method-based technique [8]. They pointed out that the composites periodically constructed by cubically shaped dielectric inclusions and hosting medium can attain the Maxwell–Garnet bound due to the strongest edge effect between adjacent particles (i.e., mutual coupling effects) [31]. Being the lowest bound for the static effective permittivity of any isotropic two-phase mixture, the Maxwell–Garnet bound is actually the lower HS bound in 3-D scenarios. To verify this, we use this new homogenization formula to compute the effective permittivity for the composites with cubic inclusion in different volume fractions. It is assumed that the permittivities of the inclusion and and . As in free space are isotropic and equal to Fig. 1, the gray (the gray and dark appears blue and red, respectively, in the color print of this paper) diamonds clearly indicate that the effective values are very close to the lower HS bound. In contrast to these points near the lower HS bound, the composites with free-space inclusions are found to have higher effective values very close to the upper HS bound (denoted by dark solid line in Fig. 1). The attainability of the HS bounds has been discussed in detail in [32] for multiphase composites with extremal conductivity. One of the main differences between our method and the method proposed by Quchetto et al. [5] is their derivative operators; the former uses the curl operator and the latter adopted the gradient operator. This is why the curl-operator-based and gradient-operator-based are named in Fig. 1. The other difference is the measurement of the local property. Unlike the gradient operator based homogenization that uses the interpolated permittivity as the local property, the curl operator-based homogenization, (16), uses the local

Fig. 1. Relation of the effective permittivity to the volume fraction of the cubeshaped inclusions hosted in free space.

matrix , which is more relevant to the inverse formulation of the local permittivity and permeability. For example, if the two compositions are isotropic and have the same permeability but different permittivity, the local values in our method should be the reciprocal of the integrated permittivity. It is worth mentioning that the conclusions drawn on the effective permittivity in this and following examples are also applicable to the effective permeability due to their interchangeable relationship. Since these kinds of composites with cubic inclusions were reported to have the lowest permittivity [8] while Fig. 1 shows that the effective values (obtained from the curl-operated-based homogenization and indicated by the blue or gray diamonds in color or black–white print, respectively) are consistently lower than those indicated by the red or dark squares (obtained from the gradient-based method in [5]), the new homogenization method seems to be more appropriate for these commonly used engineering composites. As an alternative, however, the curl-operator-based method appears to be more sophisticated and may require more computing resources than the gradient-operator-based method. Those composites in Fig. 1 with cubic inclusions were generated from a size optimization targeted for the lowest permittivity [8]. Is it possible to find other novel composite architectures that approach the lower HS bound more closely? If started from two different initial distributions, namely the relative density of Phase 1 in each element is either directly or inversely proportional to its distance to the center of the base cell, two optimal base cell configurations are obtained, as shown in Fig. 2(a) and (b), by using the inverse homogenization method. To illustrate the internal structure of the base cell, only Phase 1 (red (in online version) in color print or dark in black–white print) or Phase 2 (blue (in online version) or gray) is plotted in the following figures. Although the structures look different in a single base cell, they indeed construct the same composites if different adjacent base cells are periodically materialized in the three orthogonal directions as in Fig. 2(c). As in Fig. 2(b), Phase 1 accumulates in the center of the base cell with rounded edges and corners, and bumps up in the centers of six surfaces. The effective values for these two base cells are similar and correspond to point A in Fig. 3, which appears closer to the lower HS bound than Point B, indicating the effective


914


Fig. 2. Microstructures of the composite with the lowest isotropic permittivity: (a) the optimal base cell resulted from initial value 1, (b) the optimal base cell resulted from initial value 2, and (c) the 2 2 2 ranked optimal base cells. The targets are " = " = " = 2:0390. The attained values are " = " = " = 2:0626 and " = 2:0616, " = " = 2:0602 in (a) and (b), respectively, with the volume fraction of V = 0:3430.

2 2

Fig. 4. Microstructures of the composite with anisotropic permittivity: (a) the base cell and (b) the 4 4 4 ranking of base cells. The targets are " = 2:9475, " = 2:1446, and " = 5:6080. The attained values are " = 2:9776, " = 2:1869, and " = 5:4885. The volume fraction equals the target of V = 0:5120.

2 2

Fig. 5. Microstructures of the composite with the maximal permittivity and permeability: (a) the base cell and (b) the 2 2 2 ranking of base cells. The attained permittivity and permeability are " = " = " = 4:0771 and = = = 4:0768, respectively. The volume fraction of phase 1 equals the target V = 0:5.

2 2

Fig. 3. Microstructures of the composites with different isotropic permittivities but the same volume fraction (V = 0:3430) of phase 1.

value of the composite with the cubic inclusions obtained from the size optimization [8]. Fig. 3 also illustrates other microstructures with the effective permittivity varying from the lower to upper HS bounds. It is clearly observed that, from points A to F, the isolated inclusions gradually become interconnected to enhance the magnetic permittivity in the base cells. For the base cells corresponding to points C-E, both phases are continuous and the effective value seems to be directly related to the minimum sectional area of the inclusions. Anisotropic 3-D composites can also be designed with the inverse homogenization method. For example, if the target is located at the lower Milton–Kohn surface [33], where the third is equal to a value as high as the principal permittivity arithmetic bound [30], the dielectric material takes a cylindrical shape that has its major axis parallel to the direction of the

maximal permittivity. As the other two principal entries are not , the cross section of the cylinder may not be equal squared-symmetric as in Fig. 4. It is of significant benefit for metamaterials to generate more than one physical property by maximizing the utilization of the properties of different components. Bearing this in mind, the two components in the following example have the competing , and , . The properties as objective function becomes the summation of the reciprocals of the principal entries of the permittivity and permeability (27) and denote the weighting factors to emphasize where the relative importance of the permittivity and permeability, respectively. If the volume fractions for these two phases are equal, the optimal microstructure can be obtained as in Fig. 5. It looks very similar to the well-known Schwarz primitive structure [34], which is a class of the structures with the minimal surface area. Such a structure was also reported by Torquato et al. in [22], which attained the maximal transport properties of heat and electricity simultaneously. Since these two phases occupy the same volume in the base cell, the effective permittivity and



Fig. 6. Microstructures of the composite with the maximal permittivity and permeability: (a) the base cell and (b) the 2 2 2 ranking of base cells. The attained permittivity and permeability are " = " = " = 6:5685 and = = = 2:2296, respectively. The volume fraction of phase 1 equals the target V = 0:7499.

2 2

permeability are almost the same (4.0771 and 4.0768, respectively), and both are close to the upper HS bound (4.7059), even though their properties compete with each other. However, for the composites with an increasing volume fraction of Phase 1, as shown in Fig. 6, the effective permittivity is much larger than the effective permeability, simply because the phase with strong permittivity occupies a larger volume. Similarly to the last example, the effective permittivity (6.5685) and effective permeability (2.2296) are close to their upper HS bounds (7.0968 and 2.7027), respectively. The optimized microstructure in Fig. 6 also approximately represents another class of special structures, whose surface mean curvature is not zero but a constant anywhere. More details about these structures were reported in [35] In order to obtain negative refraction, we can substitute the normal components in the abovementioned two examples with nondispersive chiral material and resonant dipole particles because the optimal properties of these structures are independent of the components and physical properties [20]–[22], [29], [36]. Pendry [6], [9], [37] indicated that the chiral composite can attain negative refraction at the resonant frequency of the dipoles. To guarantee the isotropy of the chiral composite, these two components should be randomly mixed or symmetrically distributed within base cells which are periodically repeated in a lattice [7]. The Schwarz primitive structure and its derivatives obtained from the optimization could be ideal geometrical models for chiral composite for the following three reasons. 1) Their properties are isotropic due to the cubic-symmetry structures. 2) It has been reported that such structures can best utilize the features of different components for electric and thermal properties [22]. For example, the nonmagnetic inclusive core and metal shell could be the Schwarz-type structures with different surface mean curvatures. Similarly, other analogous structures with constant mean curvatures [29], [35] have potential to be used for chiral composites. 3) Since the geometrical characteristics of these structures can be expressed by mathematical formula (e.g., the nano-spheres [38]), it is possible to derive analytical equations for the effective properties. Recently, Qiu et al. [37] systemically studied the effective properties for special

915

Fig. 7. Microstructures of the composite with the maximal permittivity and permeability: (a) base cell and (b) 2 2 2 ranking of base cells. The attained permittivity and permeability are " = " = " = 2:4295. The volume fraction of phase 1 equals the target V = 0:3430.

2 2

chiral composite constructed by dielectric spheres covered in plasmonic shells. Similar work can be taken for the Schwarz-type and its analogous structures in the future. The solution to the inverse homogenization problem may not be unique. It has been shown in [20] that very different microstructures might have the same or close effective properties. For example, the microstructure generated in Fig. 7 has different topologies from those shown in Fig. 3, but its effective permittivity locates between the lower and upper HS bounds, which appears somewhat disagreeable with the topological transition shown in Fig. 3. A simple elucidation of the multiple solutions to the design problem might be due to the same cross-sectional area of inclusions, which largely determines the permittivity between the base cells. Nevertheless, more work about this conjecture is needed. V. CONCLUSION In accordance with the homogenization theory, the effective electromagnetic properties are estimated by considering the composition and configuration of the base cells. To design novel composites with special electromagnetic properties, the inverse homogenization technique is developed. The numerical results showed that the effective values obtained from the new homogenization formula appear closer to the bounds for the composites with inclusions hosting in free space, indicating that it may be more appropriate to be used for tailoring the effective properties of these composites. In this paper, several microstructural composites having intermediate effective values ranging from the lower to the upper HS bounds are obtained. Their topological transition indicates that the effective permeability is determined by the minimum cross-sectional area of dielectric material in a certain direction. The Schwarz primitive structures with minimal surface are also observed when seeking the composites with the maximal permeability and permittivity simultaneously. This structure and its analogous configurations can be generalized for chiral composites with negative refraction by filling them with nondispersive core and resonant dipole shell. REFERENCES [1] C. Caloz and T. Itoh, Electromagnetic Metamaterials: Transmission Line Theory and Microwave Applications. Hoboken, NJ: Wiley, 2006.


916


[2] J. B. Pendry, A. Holden, and D. Robbins et al., “Magnetism from conductors and enhanced nonlinear phenomena,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 11, pp. 2075–2084, Nov. 1999. [3] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of " and ,” Sov. Phys.–Usp., vol. 10, no. 4, pp. 509–514, 1968. [4] J. Hong and M. Lancaster, Microstrip Filters for RF/Microwave Applications. New York: Wiley, 2001. [5] O. Ouchetto, C. W. Qiu, and S. Zouhdi et al., “Homogenization of 3-D periodic bianisotropic metamaterials,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 11, pp. 3893–3898, Nov. 2006. [6] J. B. Pendry, “A chiral route to negative refraction,” Science, vol. 306, no. 5700, pp. 1353–1355, Nov. 2004. [7] S. Tretyakov, A. Sihvola, and L. Jylha, “Backward-wave regime and negative refraction in chiral composites,” Photon. Nanostruct. Fundam. Appl., vol. 3, no. 2–3, pp. 107–115, Oct. 2005. [8] K. Whites and F. Wu, “Effects of particle shape on the effective permittivity of composite materials with measurements for lattices of cubes,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1723–1729, Jul. 2002. [9] C. W. Qiu, H. Y. Yao, and L. W. Li et al., “Backward waves in magnetoelectrically chiral media: Propagation, impedance, and negative refraction,” Phys. Rev. B, Condens. Matter, vol. 75, no. 15, p. 155120, Apr. 2007. [10] C. Brosseau and A. Beroual, “Effective permittivity of composites with stratified particles,” J. Phys. D, Appl. Phys., vol. 34, no. 5, pp. 704–710, Mar. 2001. [11] A. Mejdoubi and C. Brosseau, “Numerical calculations of the intrinsic electrostatic resonances of artificial dielectric heterostructures,” J. Appl. Phys., vol. 101, pp. 084109/1–084109/13, Apr. 2007. [12] A. Bensoussan, G. Papanicolaou, and J. L. Lions, Asymptotic Analysis for Periodic Structures. Amsterdam, The Netherlands: Elsevier, 1978. [13] E. Sanchez-Palencia, Non-Homogeneous Media and Vibration Theory. Berlin, Germany: Springer, 1980. [14] M. P. Bendsøe and N. Kikuchi, “Generating optimal topologies in structural design using a homogenization method,” Comput. Methods Appl. Mech. Eng., vol. 71, no. 2, pp. 197–224, Nov. 1988. [15] O. Sigmund, “Materials with prescribed constitutive parameters—An inverse homogenization problem,” Int. J. Solids Struct., vol. 31, no. 17, pp. 2313–2329, Sep. 1994. [16] K. Svanberg, “The method of moving asymptotes—A new method for structural optimisation,” Int. J. Numer. Meth. Eng., vol. 24, no. 2, pp. 359–373, Feb. 1987. [17] E. J. Haug, K. K. Choi, and V. Komkov, Design Sensitivity Analysis of Structural Systems. Orlando, FL: Academic, 1986. [18] O. Sigmund, “Materials with prescribed constitutive parameters—An inverse homogenization problem,” Int. J. Solids Struct., vol. 31, no. 17, pp. 2313–2329, Sep. 1994. [19] O. Sigmund and S. Torquato, “Composites with extremal thermal expansion coefficients,” Appl. Phys. Lett., vol. 69, no. 21, pp. 3203–3205, Nov. 1996. [20] S. W. Zhou and Q. Li, “Computational design of multiphase microstructural materials for extremal conductivity,” Comput. Mater. Sci., vol. 43, no. 3, pp. 549–564, Mar. 2008. [21] J. K. Guest and J. H. Prévost, “Design of maximum permeability material structures,” Comput. Methods Appl. Mech. Eng., vol. 196, no. 4–6, pp. 1006–1017, Oct. 2007. [22] S. Torquato, S. Hyun, and A. Donev, “Multifunctional composites: Optimizing microstructures for simultaneous transport of heat and electricity,” Phys. Rev. Lett., vol. 89, no. 26, p. 266601, Dec. 2002. [23] Y. H. Chen, S. W. Zhou, and Q. Li, “Computational design for multifunctional microstructural composites,” J. Modern Phys. B, vol. 23, no. 6–7, pp. 1345–1351, 2009. [24] M. Y. Wang, S. Zhou, and H. Ding, “Nonlinear diffusions in topology optimisation,” Struct. Multidiscip. Opt., vol. 28, no. 4, pp. 262–276, Oct. 2004. [25] Y. M. Xie and G. P. Steven, “A simple evolutionary procedure for structural optimization,” Comput. Struct., vol. 49, no. 5, pp. 885–896, Dec. 1993. [26] G. P. Steven, Q. Li, and Y. Xie, “Evolutionary topology and shape design for mathematical physical problems,” Comput. Mech., vol. 26, no. 2, pp. 129–139, Aug. 2000. [27] J. M. Jin and V. Liepa, “Simple moment method program for computing scattering from complex cylindrical obstacles,” Proc. Inst. Elect. Eng.—Microw., Antennas Propag., vol. 136, no. 4, pp. 321–329, Aug. 1989.

[28] Z. Hashin and S. Shtrikman, “A variational approach to theory of effective magnetic permeability of multiphase materials,” J. Appl. Phys., vol. 33, no. 10, pp. 3125–3131, Oct. 1962. [29] S. W. Zhou and Q. Li, “The relation of constant mean curvature surfaces to multiphase composites with extremal thermal conductivity,” J. Phys. D, Appl. Phys., vol. 40, no. 19, pp. 6083–6093, Oct. 2007. [30] O. Wiener, “Die theorie des mischkorpers fur das feld der stationaren stromung,” Abh. Math.-Phys. Klasse Koniglich Sachsischen Des. Wiss., vol. 32, pp. 509–604, 1912. [31] J. C. Maxwell-Garnett, “Colours in metal glasses and in metallic films,” Phil. Trans. R. Soc. Lond. A, Math. Phys. Sci., vol. 203, pp. 385–420, 1904. [32] S. W. Zhou and Q. Li, “A microstructure diagram for known bounds in conductiity,” J. Mater. Res., vol. 23, no. 3, pp. 798–811, Mar. 2008. [33] G. W. Milton and R. V. Kohn, “Variational bounds on the effective moduli of anisotropic composites,” J. Mech. Phys. Solids, vol. 36, no. 6, pp. 597–629, Jun. 1988. [34] H. Schwarz, Gesammelte Mathematische Abhandlungen. Berlin, Germany: Springer, 1890. [35] W. Góz´dz´ and R. Holyst, “Triply periodic surfaces and multiply continuous structures from the Landau model of microemulsions,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 54, no. 5, pp. 5012–5027, Nov. 1996. [36] S. W. Zhou and Q. Li, “A variational level set method for the topology optimization of steady-state Navier–Stokes flow,” J. Comput. Phys., vol. 227, no. 24, pp. 10178–10195, Dec. 2008. [37] C. W. Qiu and L. Gao, “Resonant light scattering by small coated nonmagnetic spheres: Magnetic resonances, negative refraction, and prediction,” J. Opt. Soc. Amer. B, Opt. Phys., vol. 25, pp. 1728–1737, Oct. 2008. [38] M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. E, Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top., vol. 73, Jan. 2006, Art. ID 045105. Shiwei Zhou received the Ph.D. degree from The Chinese University of Hong Kong, Hong Kong, in 2006. He has been a Research Fellow with the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney, N.S.W., Australia, since 2006. His research interests include topology optimization, computational material design, metamaterials, and bioengineering.

Wei Li received the Ph.D. degree from The University of Sydney, Sydney, N.S.W., Australia, in 2002. She was an Australian Research Council (ARC) Australian Postdoctoral Fellow from 2006 to 2008 and now is an ARC Australian Research Fellow with the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney. Her recent research interests include topographical design and fabrication of micro- and nano-structured materials. She is also interested in modeling of electrical and magnetic fields in biological tissues.

Qing Li received the Ph.D. degree from The University of Sydney, Sydney, N.S.W., Australia, in 2000. He was a Postdoctoral Research Fellow with Cornell University, Ithaca, NY, from 2000 to 2001, and an Australian Research Council Australian Postdoctoral Fellow with The University of Sydney from 2001 to 2003. He was a Senior Lecturer with James Cook University, Townsville, Q.L.D., Australia, from 2004 to 2006. He has been with The University of Sydney since 2006 and is now an Associate Professor with the School of Aerospace, Mechanical and Mechatronic Engineering. His research interests include topology optimization for multiphysics and general field problems with a focus on microstructural material design for transporting properties.
